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18.034 Honors Differential Equations
Spring 2009
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18.034 Midterm #1
Name:
1. (20 points) Solve the initial value problem
y � y �� − t = 0,
y(1) = 2,
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y � (1) = 1.
18.034 Midterm #1
Name:
2. Consider the differential equation y � = y(5 − y)(y − 4)2 .
(a) (7 points) Determine the critical points (stationary solutions).
(b) (5 points) Sketch the graph of f (y) = y(5 − y)(y − 4)2 .
(c) (8 points) Discuss the stability of critical points in part (b).
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18.034 Midterm #1
Name:
3. (20 points) Determine the values of a, if any, for which all solutions of the differential equation
y �� − (3 − a)y � − 2(a − 1)y = 0
tend to zero as t → ∞. Here, � =
d
dt .
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18.034 Midterm #1
Name:
4. Consider the undamped forced vibration system
y �� + y = 3 cos ωt,
y(0) = 0,
y � (0) = 0.
(a) (10 points) Find the solution for ω �= 1.
(b) (5 points) Find the solution for ω = 1.
(c) (5 points) Discuss the behavior of solutions in part (a) and part (b).
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18.034 Midterm #1
Name:
5. (a) (5 points) State the Sturm Comparison Theorem.
(b) (15 points) Show that no nontrivial solution of y �� + (1 − t2 )y = 0 vanishes infinitely often on
0 < t < ∞.
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